Maxwell's Equations

Now, Maxwell's Equations are a concise summary of the basic laws presented above. Obviously, there are more basic laws of electromagnetism than there are Maxwell's Equation. Some basic laws are subsets of others. Coulomb's Law falls from Gauss' Law of electric fields for stationary charges. In the same manner, the Biot-Savart Law comes from Ampere's law for individual moving charges.

This leaves us with conservation of the electric field flux and of the magnetic field flux which are Gauss' Laws of electricity and magnetism, respectively. Additionally, there is the induction of an electric field due to a changing magnetic field - Faraday's Law of Induction. The complementary relationship between the creation of a magnetic field and the motion of charge - Ampere's Law. The Maxwell equations follow in the same order as listed above.

$$\begin{align} \iint_{A} \mathbf{E} \cdot \mathbf{\hat{n}} \;... ...bf{\hat{t}} \; ds &= \mu_0 i &\text{(Ampere's Law)} \end{align}$$ (19)

However, these equations are more commonly listed in differential form.

$$\begin{align} \nabla \cdot \mathbf{E} &= \frac{\rho_e}{\epsi... ...thbf{E}}{\partial t} \right) &\text{(Ampere's Law)} \end{align}$$ (20)

The equations above assume ideal conductors and vacuums amongst other things. In reality electric fields are affected by the polarization of atoms in materials and magnetic field are similarly changed by the magnetization of materials.

If $\mathbf{E}$ is the intensity of the electric field at some point, then $\mathbf{D}$ is the displacement field which is actual strength of the field due to the electric field and the displacement, or polarization, of electrons in the material due to the original electric field. Since the polarization, $\mathbf{P}$, is a result of the electric field, $\mathbf{E}$,then the displacement affects can be included into the constant, $\epsilon = \epsilon_0 + \frac{\mathbf{P}}{\mathbf{E}}$. Note that $\epsilon$ is only a scalar constant when the material's displacement properties are isotropic; otherwise, $\epsilon$ is a tensor.

$$\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P} =\epsilon \mathbf{E}$$ (21)

For magnetic fields a similar relationship exists. However, $\mathbf{B}$is the actual field strength including the affects of the magnetic field intensity and the magnetization of the material. Again, $\mu$ has the equivalent properties as $\epsilon$.

$$\mathbf{B} = \mu_0 \mathbf{H} + \mathbf{M} =\mu \mathbf{H}$$ (22)

Finally, the Maxwell Equations below include affects of electric polarization and magnetization of materials using the magnetic instensity, $\mathbf{H}$, and electric field strength, $\mathbf{D}$,as defined above.

$$\begin{align} \nabla \cdot \mathbf{D} &= \rho_e &\text{(Gaus... ...thbf{D}}{\partial t} \right) &\text{(Ampere's Law)} \end{align}$$ (23)